invited to read~\cite{BT89,bahi07}.
Before using an asynchronous iterative method, the convergence must be
-studied. Otherwise, there is no garantee that the application will reach the convergence. An
+studied. Otherwise, there is no guarantee that the application will reach the convergence. An
algorithm that supports both the synchronous or the asynchronous iteration model
requires very few modifications to be able to be executed in both variants. In
practice, only the communications management and the convergence detection are different. In
The number of iterations required to reach the convergence is generally greater
for the asynchronous scheme (this number depends on the delay of the
messages). Note that, it is not the case in the synchronous mode where the
-number of iterations is the same than in the sequential mode. In this way, the
+number of iterations is the same as in the sequential mode. Thus, the
set of the parameters of the platform (number of nodes, power of nodes,
inter and intra clusters bandwidth and latency,~\ldots) and of the
application can drastically change the number of iterations required to get the
these resources along the program execution. This model produces accurate
results while still running relatively
fast~\cite{bedaride+degomme+genaud+al.2013.toward,velho+schnorr+casanova+al.2013.validity}.
-During the simulation, the computation is really executed, but the commuications
+During the simulation, the computations are really executed, but the communications
are intercepted and their execution time evaluated according to the parameters
of the simulated platform. It is also possible for SimGrid/SMPI to only keep the
duration of large computations by skipping them. Moreover, when applicable, the
\label{sec:04}
\subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
\label{sec:04.01}
-In this paper we focus on two-stage multisplitting methods in their both versions (synchronous and asynchronous)~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$:
+In this paper we focus on two-stage multisplitting methods in both their versions (synchronous and asynchronous)~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$:
\begin{equation}
Ax=b,
\label{eq:01}
x_\ell^{k+1} = A_{\ell\ell}^{-1}(b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m}),\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots
\label{eq:02}
\end{equation}
-where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system:
+where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel so that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system:
\begin{equation}
A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
\label{eq:03}
\end{equation}
-where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
+where the right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. Line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using the GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
\begin{figure}[htpb]
%\begin{algorithm}[t]
\min_{\alpha\in\mathbb{R}^s}{\|b-AS\alpha\|_2}.
\label{eq:06}
\end{equation}
-The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
+The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using the CGLS method~\cite{Hestenes52} sosuch that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
\begin{figure}[htbp]
%\begin{algorithm}[t]
One of our objectives when simulating the application in SimGrid is, as in real
life, to get accurate results (solutions of the problem) but also to ensure the
-test reproducibility under the same conditions. According to our experience,
+test reproducibility under similar conditions. According to our experience,
very few modifications are required to adapt a MPI program for the SimGrid
simulator using SMPI (Simulated MPI). The first modification is to include SMPI
libraries and related header files (\verb+smpi.h+). The second modification is to
effects on runtime between the threads running in the same process and generated by
SimGrid to simulate the grid environment.
-\paragraph{Parameters of the simulation in SimGrid}
+\paragraph{Simulation parameters for SimGrid}
\ \\ \noindent Before running a SimGrid benchmark, many parameters for the
computation platform must be defined. For our experiments, we consider platforms
-in which several clusters are geographically distant, so there are intra and
+in which several clusters are geographically distant, so that there are intra and
inter-cluster communications. In the following, these parameters are described:
\begin{itemize}
have been chosen for the study in this paper. \\
\textbf{Step 2}: Collect the software materials needed for the experimentation.
-In our case, we have two variants algorithms for the resolution of the
-3D-Poisson problem: (1) using the classical GMRES; (2) and the multisplitting
+In our case, we have two variants for the resolution of the
+3D-Poisson problem: (1) using the classical GMRES; (2) using the multisplitting
method. In addition, the SimGrid simulator has been chosen to simulate the
behaviors of the distributed applications. SimGrid is running in a virtual
machine on a simple laptop. \\
represents the number of clusters in the grid and the second number represents
the number of hosts (processors/cores) in each cluster. \\
-\textbf{Step 5}: Conduct an extensive and comprehensive testings
+\textbf{Step 5}: Conduct extensive and comprehensive testings
within these configurations by varying the key parameters, especially
the CPU power capacity, the network parameters and also the size of the
input data. \\
is the network configuration. Two main network parameters can modify drastically
the program output results:
\begin{enumerate}
-\item the network bandwidth ($bw$ in bits/s) also known as "the data-carrying
- capacity" of the network is defined as the maximum of data that can transit
+\item the network bandwidth ($bw$ in Gbits/s) also known as "the data-carrying
+ capacity" of the network is defined as the maximum amount of data that can transit
from one point to another in a unit of time.
\item the network latency ($lat$ in microseconds) defined as the delay from the
- start time to send a simple data from a source to a destination.
+ starting time to send a simple data from a source to a destination.
\end{enumerate}
-Upon the network characteristics, another impacting factor is the volume of data exchanged between the nodes in the cluster
+Among the network characteristics, another impacting factor is the volume of data exchanged between the nodes in the cluster
and between distant clusters. This parameter is application dependent.
- In a grid environment, it is common to distinguish, on one hand, the
+ In a grid environment, it is common to distinguish, on the one hand, the
\textit{intra-network} which refers to the links between nodes within a
cluster and on the other hand, the \textit{inter-network} which is the
backbone link between clusters. In practice, these two networks have
different speeds. The intra-network generally works like a high speed
- local network with a high bandwidth and very low latency. In opposite, the
- inter-network connects clusters sometime via heterogeneous networks components
- through internet with a lower speed. The network between distant clusters
+ local network with a high bandwidth and very low latency. On the contrary, the
+ inter-network connects clusters sometimes via heterogeneous networks components the through internet with a lower speed. The network between distant clusters
might be a bottleneck for the global performance of the application.
\subsection{Comparison between GMRES and two-stage multisplitting algorithms in
synchronous mode}
In the scope of this paper, our first objective is to analyze
-when the synchronous Krylov two-stage method has better performance than the
-classical GMRES method. With a synchronous iterative method, better performance
-means a smaller number of iterations and execution time before reaching the
+when the synchronous Krylov two-stage method has better performances than the
+classical GMRES method. With a synchronous iterative method, better performances
+mean a smaller number of iterations and execution time before reaching the
convergence.
Table~\ref{tab:01} summarizes the parameters used in the different simulations:
the grid architectures (i.e. the number of clusters and the number of nodes per
cluster), the network of inter-clusters backbone links and the matrix sizes of
the 3D Poisson problem. However, for all simulations we fix the network
-parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency
+parameters of the intra-clusters links: the bandwidth $bw$=10Gbit/s and the latency
$lat=8\mu$s. In what follows, we will present the test conditions, the output
results and our comments.
\begin{tabular}{ll}
\hline
Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\
-\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\
- & $N2$: $bw$=1Gbs, $lat=50\mu$s \\
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbit/s, $lat=8\mu$s \\
+ & $N2$: $bw$=1Gbit/s, $lat=50\mu$s \\
\multirow{2}{*}{Matrix size} & $Mat1$: N$_{x}\times$N$_{y}\times$N$_{z}$=150$\times$150$\times$150\\
& $Mat2$: N$_{x}\times$N$_{y}\times$N$_{z}$=170$\times$170$\times$170 \\ \hline
\end{tabular}
a given matrix size of 170$^3$ elements, a non-variation in the number of
iterations for the classical GMRES algorithm, which is not the case of the
Krylov two-stage algorithm. In fact, with multisplitting algorithms, the number
-of splitting (in our case, it is equal to the number of clusters) influences on the
-convergence speed. The higher the number of splitting is, the slower the
+of splittings (in our case, it is equal to the number of clusters) influences on the
+convergence speed. The higher the number of splittings is, the slower the
convergence of the algorithm is (see the output results obtained from
configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs.
8$\times$8).
-The execution times between both algorithms is significant with different grid
+The execution times between both algorithms are significant with different grid
architectures. The synchronous Krylov two-stage algorithm presents better
-performances than the GMRES algorithm, even for a high number of clusters (about
-$32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can
+performances than the GMRES algorithm, even for a high number of clusters (it is about
+$32\%$ more efficient on a grid of 8$\times$8 than the GMRES). In addition, we can
observe a better sensitivity of the Krylov two-stage algorithm (compared to the
GMRES one) when scaling up the number of the processors in the computational
grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is
\end{figure}
\subsubsection{Network latency impacts on performances\\}
-Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm.
+Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbit/s to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm.
\begin{figure}[ht]
\centering
Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of
$2\times16$ processors interconnected by a network of latency $lat=50\mu$s to
-solve a 3D Poisson problem of size $150^3$. The results of increasing the
-network bandwidth from $1$Gbs to $10$Gbs show the performances improvement for
-both algorithms by reducing the execution times. However, the Krylov two-stage
+solve a 3D Poisson problem of size $150^3$. Increasing the
+network bandwidth from $1$Gbit/s to $10$Gbit/s results in improving the performances of both algorithms by reducing the execution times. However, the Krylov two-stage
algorithm presents a better performance gain in the considered bandwidth
interval with a gain of $40\%$ compared to only about $24\%$ for the classical
GMRES algorithm.
\subsubsection{Matrix size impacts on performances\\}
-In these experiments, the matrix size of the 3D Poisson problem is varied from
+In these experiments, the matrix size of the 3D Poisson problem varies from
$50^3$ to $190^3$ elements. The simulated computational grid is composed of $4$
clusters of $8$ processors each interconnected by the network $N2$ (see
Table~\ref{tab:01}). As shown in Figure~\ref{fig:05}, the execution
observe that for some problem sizes, the convergence (and thus the execution
time) of the Krylov two-stage algorithm varies quite a lot.
%This is due to the 3D partitioning of the 3D matrix of the Poisson problem.
-These findings may help a lot end users to setup the best and the optimal targeted environment for the application deployment when focusing on the problem size scale up.
+These findings may greatly help a lot end users to setup the best and the optimal targeted environment for the application deployment when focusing on the problem size scale up.
\begin{figure}[ht]
\centering
To conclude these series of experiments, with SimGrid we have been able to make
many simulations with many parameters variations. Doing all these experiments
-with a real platform is most of the time not possible or very costly. Moreover
+with a real platform is most of the time impossible or very costly. Moreover
the behavior of both GMRES and Krylov two-stage algorithms is in accordance
with larger real executions on large scale supercomputers~\cite{couturier15}.
\subsection{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms}
-The previous paragraphs put in evidence the interests to simulate the behavior
+The previous paragraphs put in evidence the interest to simulate the behavior
of the application before any deployment in a real environment. In this
section, following the same previous methodology, our goal is to compare the
efficiency of the multisplitting method in \textit{ asynchronous mode} compared with the
The interest of using an asynchronous algorithm is that there is no more
synchronization. With geographically distant clusters, this may be essential.
In this case, each processor can compute its iterations freely without any
-synchronization with the other processors. Thus, the asynchronous may
-theoretically reduce the overall execution time and can improve the algorithm
+synchronization with the other processors. Thus, an asynchronous algorithm may
+theoretically reduce the overall execution time and can also improve the algorithm
performance.
In this section, the SimGrid simulator is used to compare the behavior of the
Table~\ref{tab:03} reports the relative gains between both algorithms. It is
defined by the ratio between the execution time of GMRES and the execution time
of the multisplitting. The ratio is greater than one because the asynchronous
-multisplitting version is faster than GMRES. In average, the two-stage
-multisplitting algorithm to be more than $2.5$ times faster than the classical
+multisplitting version is faster than GMRES. On average, the two-stage
+multisplitting algorithm is more than $2.5$ times faster than the classical
GMRES. These experiments also show the relative tolerance of the multisplitting
algorithm when using a low speed network as usually observed with geographically
distant clusters through the internet.
These results are important since it is very time consuming to find optimal
configuration and deployment requirements for a given application on a given
-multi-core architecture. Finding good resource allocations policies under
+multi-core architecture. Finding good resource allocation policies under
varying CPU power, network speeds and loads is very challenging and labor
intensive. This problematic is even more difficult for the asynchronous
scheme where a small parameter variation of the execution platform and of the
application data can lead to very different numbers of iterations to reach the
-converge and so to very different execution times.
+convergeence and consequently to very different execution times.
In future works, we plan to investigate how to simulate the behavior of really
-large scale applications. For example, if we are interested to simulate the
+large scale applications. For example, if we are interested in simulating the
execution of the solvers of this paper with thousand or even dozens of thousands
of cores, it is not possible to do that with SimGrid. In fact, this tool will
-make the real computation. So we plan to focus our research on that problematic.
+make the real computation. That is why, we plan to focus our research on that problematic.
%%% fill-column: 80
%%% ispell-local-dictionary: "american"
%%% End:
+
+% LocalWords: Ramamonjisoa Ziane Khodja Laiymani Raphaël Arnaud Giersch Femto
+% LocalWords: Franche Comté Belfort GMRES multisplitting SimGrid Krylov SMPI
+% LocalWords: MPI
